Dirichlet series
| L(s) = 1 | + 2-s + 6·3-s − 4-s + 6·5-s + 6·6-s + 7·7-s + 14·9-s + 6·10-s + 8·11-s − 6·12-s − 8·13-s + 7·14-s + 36·15-s − 2·16-s − 7·17-s + 14·18-s − 11·19-s − 6·20-s + 42·21-s + 8·22-s + 19·23-s + 4·25-s − 8·26-s + 9·27-s − 7·28-s + 3·29-s + 36·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 3.46·3-s − 1/2·4-s + 2.68·5-s + 2.44·6-s + 2.64·7-s + 14/3·9-s + 1.89·10-s + 2.41·11-s − 1.73·12-s − 2.21·13-s + 1.87·14-s + 9.29·15-s − 1/2·16-s − 1.69·17-s + 3.29·18-s − 2.52·19-s − 1.34·20-s + 9.16·21-s + 1.70·22-s + 3.96·23-s + 4/5·25-s − 1.56·26-s + 1.73·27-s − 1.32·28-s + 0.557·29-s + 6.57·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{7} \cdot 67^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(7^{7} \cdot 67^{7}\) |
| Sign: | $1$ |
| Analytic conductor: | \(10331.1\) |
| Root analytic conductor: | \(1.93519\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((14,\ 7^{7} \cdot 67^{7} ,\ ( \ : [1/2]^{7} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(55.68225540\) |
| \(L(\frac12)\) | \(\approx\) | \(55.68225540\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 - T )^{7} \) |
| 67 | \( ( 1 + T )^{7} \) | |
| good | 2 | \( 1 - T + p T^{2} - 3 T^{3} + 7 T^{4} - 5 T^{5} + 7 p T^{6} - 23 T^{7} + 7 p^{2} T^{8} - 5 p^{2} T^{9} + 7 p^{3} T^{10} - 3 p^{4} T^{11} + p^{6} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 - 2 p T + 22 T^{2} - 19 p T^{3} + 119 T^{4} - 70 p T^{5} + 350 T^{6} - 590 T^{7} + 350 p T^{8} - 70 p^{3} T^{9} + 119 p^{3} T^{10} - 19 p^{5} T^{11} + 22 p^{5} T^{12} - 2 p^{7} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 - 6 T + 32 T^{2} - 107 T^{3} + 71 p T^{4} - 934 T^{5} + 102 p^{2} T^{6} - 5668 T^{7} + 102 p^{3} T^{8} - 934 p^{2} T^{9} + 71 p^{4} T^{10} - 107 p^{4} T^{11} + 32 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 - 8 T + 70 T^{2} - 445 T^{3} + 2320 T^{4} - 10796 T^{5} + 43499 T^{6} - 151134 T^{7} + 43499 p T^{8} - 10796 p^{2} T^{9} + 2320 p^{3} T^{10} - 445 p^{4} T^{11} + 70 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + 8 T + 92 T^{2} + 511 T^{3} + 3463 T^{4} + 14766 T^{5} + 73070 T^{6} + 245956 T^{7} + 73070 p T^{8} + 14766 p^{2} T^{9} + 3463 p^{3} T^{10} + 511 p^{4} T^{11} + 92 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 7 T + 87 T^{2} + 359 T^{3} + 2871 T^{4} + 8905 T^{5} + 63889 T^{6} + 171282 T^{7} + 63889 p T^{8} + 8905 p^{2} T^{9} + 2871 p^{3} T^{10} + 359 p^{4} T^{11} + 87 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 11 T + 151 T^{2} + 1127 T^{3} + 8953 T^{4} + 50121 T^{5} + 15091 p T^{6} + 1242202 T^{7} + 15091 p^{2} T^{8} + 50121 p^{2} T^{9} + 8953 p^{3} T^{10} + 1127 p^{4} T^{11} + 151 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 - 19 T + 233 T^{2} - 2042 T^{3} + 14933 T^{4} - 93085 T^{5} + 520765 T^{6} - 2610828 T^{7} + 520765 p T^{8} - 93085 p^{2} T^{9} + 14933 p^{3} T^{10} - 2042 p^{4} T^{11} + 233 p^{5} T^{12} - 19 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 - 3 T + 138 T^{2} - 336 T^{3} + 9303 T^{4} - 655 p T^{5} + 397534 T^{6} - 23552 p T^{7} + 397534 p T^{8} - 655 p^{3} T^{9} + 9303 p^{3} T^{10} - 336 p^{4} T^{11} + 138 p^{5} T^{12} - 3 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 - 18 T + 237 T^{2} - 1646 T^{3} + 7524 T^{4} + 10926 T^{5} - 327828 T^{6} + 2817972 T^{7} - 327828 p T^{8} + 10926 p^{2} T^{9} + 7524 p^{3} T^{10} - 1646 p^{4} T^{11} + 237 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 20 T + 288 T^{2} - 2883 T^{3} + 24801 T^{4} - 176630 T^{5} + 1189678 T^{6} - 7272502 T^{7} + 1189678 p T^{8} - 176630 p^{2} T^{9} + 24801 p^{3} T^{10} - 2883 p^{4} T^{11} + 288 p^{5} T^{12} - 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 20 T + 418 T^{2} + 5105 T^{3} + 60797 T^{4} + 528674 T^{5} + 4438974 T^{6} + 28941876 T^{7} + 4438974 p T^{8} + 528674 p^{2} T^{9} + 60797 p^{3} T^{10} + 5105 p^{4} T^{11} + 418 p^{5} T^{12} + 20 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 + 11 T + 179 T^{2} + 1513 T^{3} + 17703 T^{4} + 127765 T^{5} + 1113473 T^{6} + 6522302 T^{7} + 1113473 p T^{8} + 127765 p^{2} T^{9} + 17703 p^{3} T^{10} + 1513 p^{4} T^{11} + 179 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 - 24 T + 398 T^{2} - 4909 T^{3} + 51748 T^{4} - 466006 T^{5} + 3779845 T^{6} - 27135350 T^{7} + 3779845 p T^{8} - 466006 p^{2} T^{9} + 51748 p^{3} T^{10} - 4909 p^{4} T^{11} + 398 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 36 T + 810 T^{2} - 13035 T^{3} + 169052 T^{4} - 34112 p T^{5} + 16522189 T^{6} - 129197874 T^{7} + 16522189 p T^{8} - 34112 p^{3} T^{9} + 169052 p^{3} T^{10} - 13035 p^{4} T^{11} + 810 p^{5} T^{12} - 36 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 - 26 T + 524 T^{2} - 7511 T^{3} + 95062 T^{4} - 994352 T^{5} + 9344229 T^{6} - 75340546 T^{7} + 9344229 p T^{8} - 994352 p^{2} T^{9} + 95062 p^{3} T^{10} - 7511 p^{4} T^{11} + 524 p^{5} T^{12} - 26 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 8 T + 358 T^{2} - 2419 T^{3} + 57614 T^{4} - 327862 T^{5} + 5474777 T^{6} - 25627946 T^{7} + 5474777 p T^{8} - 327862 p^{2} T^{9} + 57614 p^{3} T^{10} - 2419 p^{4} T^{11} + 358 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 - 16 T + 424 T^{2} - 5007 T^{3} + 80075 T^{4} - 752280 T^{5} + 8915128 T^{6} - 67475714 T^{7} + 8915128 p T^{8} - 752280 p^{2} T^{9} + 80075 p^{3} T^{10} - 5007 p^{4} T^{11} + 424 p^{5} T^{12} - 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 37 T + 905 T^{2} + 16141 T^{3} + 236871 T^{4} + 2895951 T^{5} + 30641311 T^{6} + 279737630 T^{7} + 30641311 p T^{8} + 2895951 p^{2} T^{9} + 236871 p^{3} T^{10} + 16141 p^{4} T^{11} + 905 p^{5} T^{12} + 37 p^{6} T^{13} + p^{7} T^{14} \) | |
| 79 | \( 1 + 30 T + 665 T^{2} + 11560 T^{3} + 169037 T^{4} + 2073378 T^{5} + 22512885 T^{6} + 213034480 T^{7} + 22512885 p T^{8} + 2073378 p^{2} T^{9} + 169037 p^{3} T^{10} + 11560 p^{4} T^{11} + 665 p^{5} T^{12} + 30 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 - 6 T + 425 T^{2} - 3002 T^{3} + 84089 T^{4} - 615658 T^{5} + 10305305 T^{6} - 67700620 T^{7} + 10305305 p T^{8} - 615658 p^{2} T^{9} + 84089 p^{3} T^{10} - 3002 p^{4} T^{11} + 425 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 - 18 T + 214 T^{2} - 2033 T^{3} + 26654 T^{4} - 304846 T^{5} + 2296243 T^{6} - 17604478 T^{7} + 2296243 p T^{8} - 304846 p^{2} T^{9} + 26654 p^{3} T^{10} - 2033 p^{4} T^{11} + 214 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - T + 543 T^{2} + 29 T^{3} + 133479 T^{4} + 84373 T^{5} + 19693427 T^{6} + 13999858 T^{7} + 19693427 p T^{8} + 84373 p^{2} T^{9} + 133479 p^{3} T^{10} + 29 p^{4} T^{11} + 543 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.29399860142254632853690363367, −5.18586412594227694420858502482, −5.13316963768104070401568845029, −4.55666660065647310739347352228, −4.51317453692379830171158651023, −4.47379285273681129934263033249, −4.46592602286734878418136327908, −4.15771615668160071765564743142, −4.14196772804335112361262825630, −4.12320079021775258334835464389, −3.84410545899683389839006748586, −3.38520119201803142418805368388, −3.32719291072095297096017608427, −2.96379244124993340403202134544, −2.81136784525714690380173490667, −2.64613190981081676802662072339, −2.51766330953114128796118416551, −2.25888510877070497684131040277, −2.22313886899647541799721986060, −2.14910044176243294902284112467, −2.05666710270796657426452628785, −1.65964790718091682114759036673, −1.43333236811766658809963202211, −0.994430896299902822114721843224, −0.887940221365477837367398308778, 0.887940221365477837367398308778, 0.994430896299902822114721843224, 1.43333236811766658809963202211, 1.65964790718091682114759036673, 2.05666710270796657426452628785, 2.14910044176243294902284112467, 2.22313886899647541799721986060, 2.25888510877070497684131040277, 2.51766330953114128796118416551, 2.64613190981081676802662072339, 2.81136784525714690380173490667, 2.96379244124993340403202134544, 3.32719291072095297096017608427, 3.38520119201803142418805368388, 3.84410545899683389839006748586, 4.12320079021775258334835464389, 4.14196772804335112361262825630, 4.15771615668160071765564743142, 4.46592602286734878418136327908, 4.47379285273681129934263033249, 4.51317453692379830171158651023, 4.55666660065647310739347352228, 5.13316963768104070401568845029, 5.18586412594227694420858502482, 5.29399860142254632853690363367
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.